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Properties

Label 2-588-12.11-c1-0-11
Degree $2$
Conductor $588$
Sign $-i$
Analytic cond. $4.69520$
Root an. cond. $2.16684$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·2-s + 1.73i·3-s + 2.00·4-s − 2.44i·5-s − 2.44i·6-s − 2.82·8-s − 2.99·9-s + 3.46i·10-s − 1.41·11-s + 3.46i·12-s + 4.24·15-s + 4.00·16-s + 7.34i·17-s + 4.24·18-s + 6.92i·19-s − 4.89i·20-s + ⋯
L(s)  = 1  − 1.00·2-s + 0.999i·3-s + 1.00·4-s − 1.09i·5-s − 0.999i·6-s − 1.00·8-s − 0.999·9-s + 1.09i·10-s − 0.426·11-s + 1.00i·12-s + 1.09·15-s + 1.00·16-s + 1.78i·17-s + 0.999·18-s + 1.58i·19-s − 1.09i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.546366 + 0.546366i\)
\(L(\frac12)\) \(\approx\) \(0.546366 + 0.546366i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
3 \( 1 - 1.73iT \)
7 \( 1 \)
good5 \( 1 + 2.44iT - 5T^{2} \)
11 \( 1 + 1.41T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 7.34iT - 17T^{2} \)
19 \( 1 - 6.92iT - 19T^{2} \)
23 \( 1 - 7.07T + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 3.46iT - 31T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 - 12.2iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 15.5T + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 2.44iT - 89T^{2} \)
97 \( 1 + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.61502996740223191411497840211, −10.01415601251523107256435662214, −9.138107606710670765406340393854, −8.435981689359520223731747447924, −7.87858265583587944814358275896, −6.28569130176811114730315284793, −5.47206501679519282757736022480, −4.33054233736803371382917865998, −3.09634045100185249976942965042, −1.39539794708926188978057750934, 0.63480638544888249622593666346, 2.50881962068903131585823428344, 2.93148860343749362180050577405, 5.22363794937387735565565725338, 6.42474364745786951456166727698, 7.22450911366322439824619207674, 7.43977717326687641362054378428, 8.783397814283268484463251386265, 9.420037188276493188140261151941, 10.65215547476042210772991151326

Graph of the $Z$-function along the critical line